I am going to present a theory of noncommutative cross-ratio, Schwarz derivative and their connections and relations to the operator cross-ratio. I will apply the theory to ”noncommutative elementary geometry”, relate it to noncommutative integrable systems, and provide a noncommutative version of the celebrated ”pentagramma mirificum”. Partly based on joint work with V. Shander, V. Rubtsov and G. Sharygin.